Average Error: 14.6 → 0.3
Time: 5.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2} \cdot \frac{1}{b + a}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\pi}{2} \cdot \frac{1}{b + a}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r47249 = atan2(1.0, 0.0);
        double r47250 = 2.0;
        double r47251 = r47249 / r47250;
        double r47252 = 1.0;
        double r47253 = b;
        double r47254 = r47253 * r47253;
        double r47255 = a;
        double r47256 = r47255 * r47255;
        double r47257 = r47254 - r47256;
        double r47258 = r47252 / r47257;
        double r47259 = r47251 * r47258;
        double r47260 = r47252 / r47255;
        double r47261 = r47252 / r47253;
        double r47262 = r47260 - r47261;
        double r47263 = r47259 * r47262;
        return r47263;
}
↓
double f(double a, double b) {
        double r47264 = atan2(1.0, 0.0);
        double r47265 = 2.0;
        double r47266 = r47264 / r47265;
        double r47267 = 1.0;
        double r47268 = b;
        double r47269 = a;
        double r47270 = r47268 + r47269;
        double r47271 = r47267 / r47270;
        double r47272 = r47266 * r47271;
        double r47273 = r47268 - r47269;
        double r47274 = r47267 / r47269;
        double r47275 = r47267 / r47268;
        double r47276 = r47274 - r47275;
        double r47277 = r47273 / r47276;
        double r47278 = r47272 / r47277;
        return r47278;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.6

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  2. Using strategy rm

  3. Applied difference-of-squares9.6

    \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  4. Applied associate-/r*9.1

    \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{b + a}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  5. Using strategy rm

  6. Applied associate-*r/9.1

    \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{1}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  7. Applied associate-*l/0.3

    \[\leadsto \color{blue}{\frac{\left(\frac{\pi}{2} \cdot \frac{1}{b + a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}}\]
  8. Using strategy rm

  9. Applied associate-/l*0.3

    \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{1}{b + a}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}}\]

  10. Final simplification0.3

    \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{1}{b + a}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))